Geometric detection of hierarchical backbones in real networks

Abstract

Hierarchies permeate the structure of real networks, whose nodes can be ranked according to different features. However, networks are far from treelike structures and the detection of hierarchical ordering remains a challenge, hindered by the small-world property and the presence of a large number of cycles, in particular clustering. Here, we use geometric representations of undirected networks to achieve an enriched interpretation of hierarchy that integrates features defining the popularity of nodes and similarity between them, such that the more similar a node is to a less popular neighbor the higher the hierarchical load of the relationship. The geometric approach allows us to measure the local contribution of nodes and links to the hierarchy within a unified framework. Additionally, we propose a link filtering method, the similarity filter, able to extract hierarchical backbones containing the links that represent statistically significant deviations with respect to the maximum entropy null model for geometric heterogeneous networks. We applied our geometric approach to the detection of similarity backbones of real networks in different domains and found that the backbones preserve local topological features at all scales. Interestingly, we also found that similarity backbones favor cooperation in evolutionary dynamics modeling social dilemmas.

Self-Similar Growth and Synergistic Link Prediction in Technology-Convergence Networks: The Case of Intelligent Transportation Systems

“…For a time-evolving network, if its growth mechanism approximates the inverse process of the box-covering algorithm, the snapshot at time t − 1 could be considered as a scaled-down replica of the snapshot at time t. As a result, we could observe self-similar behaviors of the network statistics on the rescaled snapshots. Commonly adopted empirical evidence for the self-similar growth is the curve overlapping of rescaled network statistics, including rescaled degree distribution, rescaled clustering coefficient, rescaled degree-degree correlation, and the community structure [28,[33][34][35][36][37]. Following the convention in the literature, this study provides empirical evidence for the self-similar growth of the time-evolving technology-convergence network based on the aforementioned rescaled network statistics.…”

“…From the viewpoint of static topological structure, typical properties of fractal networks include self-similarity and scale invariance [29][30][31][32]. Using historical patent data in the ITS field, this study first provides empirical evidence for the self-similar growth of the time-evolving technology-convergence network based on the commonly adopted rescaled network statistics [28,[33][34][35][36][37] and then verifies the fractality of each snapshot using a community-structure-based approximation of the box-cover algorithm [38].…”

“…The self-similar growth of a time-evolving network is characterized by the fact that its topological property remains in a steady state while its average degree increases moderately over time [28]. Based on the analysis framework widely adopted in the previous literature [28,[33][34][35][36][37], this study first illustrates the evolution of the average degree in G hist , and then demonstrates the stability of five topological properties: (1) complementary cumulative distribution of the rescaled degree P c (d res ); (2) clustering coefficient over the rescaled degree C(d res ); (3) normalized average degree of neighbors over the rescaled degree d res nn (d res ); (4) modularity Q of the community partitions by the Louvain method;…”

“…This subsection presents the empirical evidence demonstrating the self-similar growth and fractality of the technology-convergence network. According to the existing literature [28,[33][34][35][36][37], the main criterion for the self-similar growth of a time-evolving network is that its size increases moderately over time while its topological property remains in a steady state. In terms of the growth of network sizes, Figure 5 shows that the number of nodes and the average weighted degree of the historically cumulative IPC co-occurrence network G hist moderately increase over time.…”